Intervals without primes near an iterated linear recurrence sequence

Let Formula: see text be a fixed positive integer. Let Formula: see text be a linear recurrence sequence for every Formula: see text, and we set Formula: see text, where Formula: see text. In this paper, we obtain sufficient conditions on Formula: see text so that the intervals Formula: see text do not contain any prime numbers for infinitely many integers Formula: see text, where Formula: see text is an explicit positive constant depending only on the orders of Formula: see text. As a corollary, we show that if for each Formula: see text, the sequence Formula: see text is positive, strictly increasing, and the constant term of its characteristic polynomial is Formula: see text, then for every Pisot or Salem number Formula: see text, the numbers Formula: see text are composite for infinitely many integers Formula: see text.